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Theorem wpthswwlks2on 29253

Description: For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.) (Revised by AV, 16-Mar-2022.)

**Assertion**
Ref	Expression
wpthswwlks2on	⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Proof of Theorem wpthswwlks2on Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlknon 29149	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵))
2	1	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
3	2	anbi1d 630	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4		3anass 1095	. . . . . . 7 ⊢ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
5	4	anbi1i 624	. . . . . 6 ⊢ (((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
6		anass 469	. . . . . 6 ⊢ (((𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
7	5, 6	bitri 274	. . . . 5 ⊢ (((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
8	3, 7	bitrdi 286	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))))
9	8	rabbidva2 3434	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)})
10		usgrupgr 28480	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
11		wlklnwwlknupgr 29178	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalksN 𝐺)))
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalksN 𝐺)))
13	12	bicomd 222	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
14	13	adantr 481	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
15		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(Walks‘𝐺)𝑤)
16		simprl 769	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘0) = 𝐴)
17	16	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘0) = 𝐴)
18		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑓) = 2 → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
19	18	ad2antll 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
20		simprr 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘2) = 𝐵)
21	20	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘2) = 𝐵)
22	19, 21	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = 𝐵)
23		eqid 2732	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24	23	wlkp 28911	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓(Walks‘𝐺)𝑤 → 𝑤:(0...(♯‘𝑓))⟶(Vtx‘𝐺))
25		oveq2 7419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑓) = 2 → (0...(♯‘𝑓)) = (0...2))
26	25	feq2d 6703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑓) = 2 → (𝑤:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ↔ 𝑤:(0...2)⟶(Vtx‘𝐺)))
27	24, 26	syl5ibcom 244	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(Walks‘𝐺)𝑤 → ((♯‘𝑓) = 2 → 𝑤:(0...2)⟶(Vtx‘𝐺)))
28	27	imp 407	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → 𝑤:(0...2)⟶(Vtx‘𝐺))
29		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → 𝑤:(0...2)⟶(Vtx‘𝐺))
30		2nn0 12491	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℕ₀
31		0elfz 13600	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2 ∈ ℕ₀ → 0 ∈ (0...2))
32	30, 31	mp1i 13	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → 0 ∈ (0...2))
33	29, 32	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → (𝑤‘0) ∈ (Vtx‘𝐺))
34		nn0fz0 13601	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 ∈ ℕ₀ ↔ 2 ∈ (0...2))
35	30, 34	mpbi 229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ (0...2)
36	35	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → 2 ∈ (0...2))
37	29, 36	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → (𝑤‘2) ∈ (Vtx‘𝐺))
38	33, 37	jca 512	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → ((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)))
39	28, 38	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)))
40		eleq1 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤‘0) = 𝐴 → ((𝑤‘0) ∈ (Vtx‘𝐺) ↔ 𝐴 ∈ (Vtx‘𝐺)))
41		eleq1 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤‘2) = 𝐵 → ((𝑤‘2) ∈ (Vtx‘𝐺) ↔ 𝐵 ∈ (Vtx‘𝐺)))
42	40, 41	bi2anan9 637	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → (((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
43	39, 42	imbitrid 243	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
44	43	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
45	44	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
46		vex 3478	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
47		vex 3478	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
48	46, 47	pm3.2i 471	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ V ∧ 𝑤 ∈ V)
49	23	iswlkon 28952	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝑓 ∈ V ∧ 𝑤 ∈ V)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(♯‘𝑓)) = 𝐵)))
50	45, 48, 49	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(♯‘𝑓)) = 𝐵)))
51	15, 17, 22, 50	mpbir3and 1342	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤)
52		simplll 773	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐺 ∈ USGraph)
53		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (♯‘𝑓) = 2)
54		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐴 ≠ 𝐵)
55		usgr2wlkspth 29054	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑓) = 2 ∧ 𝐴 ≠ 𝐵) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
56	52, 53, 54, 55	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
57	51, 56	mpbid 231	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
58	57	ex 413	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
59	58	eximdv 1920	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
60	59	ex 413	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
61	60	com23 86	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
62	14, 61	sylbid 239	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
63	62	imp 407	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
64	63	pm4.71d 562	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
65	64	bicomd 222	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → ((((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
66	65	rabbidva 3439	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → {𝑤 ∈ (2 WWalksN 𝐺) ∣ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
67	9, 66	eqtrd 2772	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
68	23	iswspthsnon 29148	. 2 ⊢ (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}
69	23	iswwlksnon 29145	. 2 ⊢ (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)}
70	67, 68, 69	3eqtr4g 2797	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 {crab 3432 Vcvv 3474 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 0cc0 11112 2c2 12269 ℕ₀cn0 12474 ...cfz 13486 ♯chash 14292 Vtxcvtx 28294 UPGraphcupgr 28378 USGraphcusgr 28447 Walkscwlks 28891 WalksOncwlkson 28892 SPathsOncspthson 29010 WWalksN cwwlksn 29118 WWalksNOn cwwlksnon 29119 WSPathsNOn cwwspthsnon 29121

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-ac 10113 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-xnn0 12547 df-z 12561 df-uz 12825 df-fz 13487 df-fzo 13630 df-hash 14293 df-word 14467 df-concat 14523 df-s1 14548 df-s2 14801 df-s3 14802 df-edg 28346 df-uhgr 28356 df-upgr 28380 df-umgr 28381 df-uspgr 28448 df-usgr 28449 df-wlks 28894 df-wlkson 28895 df-trls 28987 df-trlson 28988 df-pths 29011 df-spths 29012 df-pthson 29013 df-spthson 29014 df-wwlks 29122 df-wwlksn 29123 df-wwlksnon 29124 df-wspthsnon 29126

This theorem is referenced by: usgr2wspthons3 29256 frgr2wsp1 29621

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